Consider first the function
f = @(x) (x-1).*(x-2);
At the root 
, we have 
. If the values of f were perturbed at any point by noise of size, say, 
, we can imagine finding the root of the function as though drawn with a thick line, whose edges we show here.
, we have . If the values of f were perturbed at any point by noise of size, say, , we can imagine finding the root of the function as though drawn with a thick line, whose edges we show here.interval = [0.8 1.2];
fplot(f,interval), grid on, hold on
fplot(@(x) f(x)+0.02,interval,'k')
fplot(@(x) f(x)-0.02,interval,'k')
axis equal % ignore this line
xlabel('x'), ylabel('f(x)') % ignore this line
title('Well-conditioned root') % ignore this line
The possible values for a perturbed root all lie within the interval where the black lines intersect the x axis. The width of that zone is about the same as the vertical distance between the lines.
By contrast, consider the function
f = @(x) (x-1).*(x-1.01);
Now 
, and the graph of f will be much shallower near 
. Look at the effect this has on our thick rendering:
, and the graph of f will be much shallower near . Look at the effect this has on our thick rendering:axis(axis), cla
fplot(f,interval)
fplot(@(x) f(x)+0.02,interval,'k')
fplot(@(x) f(x)-0.02,interval,'k')
title('Poorly conditioned root') % ignore this line
The vertical displacements in this picture are exactly the same as before. But the potential horizontal displacement of the root is much wider. In fact, if we perturb the function upward by the amount drawn here, the root disappears entirely!